Online Calculus I Tutoring

A zero derivative does not prove a maximum or minimum

Consider the function $f(x)=x^3$, where $x$ is real. Its derivative $f^{\prime}(x)$ measures its local rate of change. At $x=0$, that derivative is zero. Does the function have a local maximum or minimum there?

Work through the reasoning

Differentiate to get $f^{\prime}(x)=3x^2$. This derivative is positive for both negative and positive nonzero $x$, so the function increases on each side of zero. It passes through $(0,0)$ without turning. Therefore zero is neither a local maximum nor a local minimum. A zero derivative identifies a point to investigate; the change in sign of the derivative tells us whether a differentiable function turns there. In an optimization problem, also check relevant endpoints and points where the function is defined but not differentiable.

Prepare for your lesson

Bring a curve-sketching or optimization question, even if you have not started. If you have work to review, show the derivative calculation separately from the sign analysis. Your tutor can help you explain what happens on either side of each candidate point before deciding whether it is an extremum.

An original Nablio example, not a university exam question. Your current course outline determines the material to focus on.